Theorem sqrtval 15290

Description: Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)

Assertion

Ref	Expression
sqrtval	⊢ (𝐴 ∈ ℂ → (√‘𝐴) = (℩𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem sqrtval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2775	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴))
2	1	3anbi1d 1468	. . 3 ⊢ (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
3	2	riotabidv 7371	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℂ ((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) = (℩𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
4		df-sqrt 15288	. 2 ⊢ √ = (𝑦 ∈ ℂ ↦ (℩𝑥 ∈ ℂ ((𝑥↑2) = 𝑦 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))
5		riotaex 7373	. 2 ⊢ (℩𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)) ∈ V
6	3, 4, 5	fvmpt 6991	1 ⊢ (𝐴 ∈ ℂ → (√‘𝐴) = (℩𝑥 ∈ ℂ ((𝑥↑2) = 𝐴 ∧ 0 ≤ (ℜ‘𝑥) ∧ (i · 𝑥) ∉ ℝ+)))

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∉ wnel 3064   class class class wbr 5110  ‘cfv 6538  ℩crio 7368  (class class class)co 7412  ℂcc 11099  0cc0 11101  ici 11103   · cmul 11106   ≤ cle 11245  2c2 12296  ℝ⁺crp 13017  ↑cexp 14099  ℜcre 15150  √csqrt 15286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-riota 7369  df-sqrt 15288

This theorem is referenced by:  sqrt0  15294  resqrtcl  15306  resqrtthlem  15307  sqrtneg  15320  sqrtcl  15415  sqrtthlem  15416